2. If 30 yards of cloth 5 quarters broad, be required to hang a bed, how many yards 3 quarters broad, will serve the same purpose? Qrs. Qrs. Yds. Or thus: As 5: 30 :: 3 5 3)150 50 yards, Ans. RULE OF FIVE DIRECT, Teaches from five given numbers to find a sixth, which must bear such proportion to the 4th and 5th, as the third bears to the 1st and 2d. GENERAL RULE. State the question so that the numbers may stand in one continued rank; and in such order that the first and fourth numbers may be of one denomination, and likewise the second and fifth. Then multiply the two first numbers together for a divisor, and the three last together for a dividend, and the quotient will be the answer, in the same name with the middle or third number. For trial of which, take the following. EXAMPLES. 1. If £100, in 12 months, gain £6, what will £25 gain in 4 months? £ P M £m £P M As 1-0-0 1·2: : 6: 2.5: : ·4 ; Here the question being stated according to rule, I find that 100, the first term, can be cancelled by 25-4, and 12, the second, by 6=2 then 4, the fifth term, is destroyed by 4, first found, and leaves 2 for a divisor and 1 for the dividend, equal to 10s. Answer. 2. If 750 bushels of oats serve 500 horse 16 days, how many bushels will serve 1000 horses 14 days? 212 RULE OF SEVEN, NINE, ELEVEN, &c. RULE OF FIVE INVERSE. This rule also may be performed by the compound rule of five num bers; observing to make the second and fourth numb ers of like name, and the third and fifth of the same, also. Then multiply the third and fourth numbers together for a divisor, and the other three for a dividend and the quotient will be the auswer, in the same name with the first number. EXAMPLES. 1. If £100, in 12 months, gain £6, what principal will gain £18 in 15 months ? Here I find 6, the third number, is contained in 18, the fifth number, 3 times; therefore I cancel 6 and 18, and set 3 under 18; then I see this 3 cancels or reduces 15,equal 5, and 5 reduces 100,equal 20; which multiplied by 12, the second number, gives 240 for the answer. 2. If 50 pioneers in 6 days cast a trench 30 yards long, how many pioneers will cast a trench 200 yards long in 3 days? p. ds. yds. ds. yds. As 5.0-6-3·0—3—200 X2=400 3)2000 Answer, 666 THE RULE OF SEVEN, NİNE, ELEVEN, &c. Teaches from seven numbers given to find an eighth, or from nine numbers given to find a tenth, or from eleven numbers given to find a twelfth, &c. This rule easily and naturally admits of sub divisions, which, from the number of the terms given, may be denominated the rule of five, the rule of seven, the rule of nine, the rule of eleven, &c. and are resolved into two parts, viz. a supposition and a demand. N. B. If five terms be given, three of these are always found in the supposition, and two in the demand; if seven terms be given,four of these are in the supposition, and three in the demand; if nine terms be given, five of these are in the supposition, and four in the demand; if eleven terms be given, six of these are in the supposition and five in the demand, &c. EXAMPLES. 1. If 15 men eat 156d. worth of bread in 6 days, when wheat is sold at 12s. per bushel, in how many days will 30 men eat 520d. worth of bread, when wheat is at 10s. per bushel? 12 days, answer. EXPLANATION. This question belongs to the rule of seven, the number sought is days, and the term of the same kind in the suppo DOUBLE RULE OF THREE IN GENERAL. 213 sition is 6 days, which I place in the middle: the remaining 6 terms ar eextremes, which I class into similar pairs, by taking one term of each pair out of the supposition, and another of the same kind out of the demand, and proceed to contract as before. 2. If 18 men build a wall 40 feet long, 3 feet thick, and 16 feet high, in 12 days, how many men will build a wall 360 feet long, 8 feet thick and 10 feet high in 60 days? This question belongs to the rule of nine, and is stated, &c. in the same manner with the preceding question in the rule of seven as follows; 3. If 12 men cast a ditch 30 feet Jong, 6 feet deep, and 3 feet broad, in 15 days, when the days are 12 hours long, in how many days will 60 men cast a ditch 300 feet lòng, 8 deep and 6 broad, when the days are but 8 hours long? 120 days, Ans. This question belongs to the rule of eleven, and is stated and wrought in the same manner with the preceding questions in the rules of seven and nine, as annexed. THE RULE OF THREE INVERSE IN VULGAR FRACTIONS. Question 1. If yard of cloth that is 2 yards wide, will make a garment, how much of any other cloth that is 3 yard wide, will make the same garment ? B. L. N. B. Here I multiply the first and second terms together, and divide by the third, the quotient is the answer. This admits of contraction, the same as before directed, as appears in the above example, viz. 3, the numerator of the third term, destroys 3 the numerator of the second, and 2, the numerator of the first, is contained in 4, the denominator of the second, 2 times; and this 2 divided into 5 the denominator of the third, gives 24 for the answer required. 2. If I lend my friend 48 dollars for of year, how much ought he to lend me for of a year? Year. : Year. រ 3 $86,40 Aus. 214 DOUBLE RULE OF THREE IN GENERAL. DOUBLE RULE OF THREE IN GENERAL, Rightly called compound proportion, because there are from five to seven numbers compounded into two, to find out a third, and these numbers ought to be placed in their right order before compounded. And no person should do business in this rule, before he partly understands to state the questions, and they are very plain for a scholar, as I shall show the work of almost every question. The double rule of three, sometimes called compound proportion, teaches by having five numbers given to find a sixth, which if the proportion be direct, must bear the same proportion to the fourth and fifth, as the third number does to the first and second; but if the proportion be inverse, the sixth number must bear the same proportion to the fourth and fifth, as the first does to the second and third. RULE. 1. State the question, by placing the three conditional terms in such order that the number which is the cause of gain, loss or action, may possess the first place; that which denotes spaces of time or distance of place, the second; and that which is the gain, loss, or action the third. 2. Place the other two terms, which move the question under those of the same name. 3. Then if the blank place, or term sought, fall under the third place, the proportion is direct, therefore multiply the three last terms together, for a dividend, and the other two for a divisor; then the quotient will be the answer. 4. But if the blank fall under the first or second place, the proportion is inverse, wherefore multiply the first, second and last terms together for a dividend, and the other two for a divisor, the quotient will be the answer. N. B. The learner may take his choice as to the mode of working the questions. 2. I make no difference in stating the questions but what is commonly used, only when the numbers are formed in their right order to shorten the operation, I divide and subdivide by any number that is easiest; first, any number that belongs to the divisor; second, any number that belongs to the dividend, by the same number, and so on throughout all the numbers that can be divided, and if there be any number, the numbers that are to form the divisor and the same number in any number that is to form the dividend, both numbers are to be cast out and if any numbers remain,they are in the same proportion as they were at first and must be worked in the same manner. Pounds are reduced to dollars by multiplying by 10 and dividing the product by 3, and the contrary. 1. Pattern. If 100 dollars gain 6 dollars in 12 months, what will 400 3. If £100 gain £6 in 12 months, how many dollars will 400 dollars gain in 9 months? £. M. £. D. M. 1,00 3)12(4 6( 9(3 10 6 6(2 8 2 Ans. 16 dollars. You seem to have a short mode in doing business, be so good as to explain it.. I will, as plain as possible, which will suffice for the whole. £100 is the principal hired, 12 months is the time hired. So I place the 12 under 100 and the blank or demand falls under the third term 6, which I place under the 400, which is the quotient, and 9 which is the time under it also, then 3 under the 100, and 12 and 10 under 400, 6, & 9 for a dividend, then I cut off the two cyphers of each, and there remains 1 and 4, then 3 divided into 12, reduces to 4, which destroys the 4 in the dividend, then 3 destroys 9, equal to 3, and 3 stands in the divisor, which destroys this 3, and there remains only 10, and 6 in the dividend, and nothing opposite in the divisor, therefore I multiply 10 by 6, equal to 60 dollars for the an swer required. Ans. 60 dolls. 4. Again, if £400 gain £18 in 9 months, how many dollars will £100 gain in 12 months? £. M. £. £. M. 4,00 9-18: 1,00 3)9(3 12 3)18(6 3)12(4 106 Here I cut off the two cyphers of the 400 and 100, there remains 1 and 4, I divide 9 and 18 by 3, which reduces them, equal to 3 and 6, and 12=4; 4 in the divisor destroys 4 in the dividend, and leaves 10 and 6 in the dividend, 60, which I divide by 3 in the diwhich multiplied together, produce 20, Ans. visor, leaves 20 for the answer. THE COMPOUND RULE OF THREE, IN VUL 3)60 GAR FRACTIONS. Question 1. If 2 acre of grass be cut down by two men in 3 day, how many acres shall be cut down by 6 men in 3 days? Men. Ds. Acres. Men. Men. Answer, 111 acres. |